I'm not familiar with that particular biography however, I can provide some background around the family's insistence. Going to Harvard wasn't necessarily about the quality of the education, per se. They provided a classics education that a young man could get at most of the colony colleges. Harvard, though, offered two key things that likely appealed to the Adams family - first, prestige and second, opportunities for networking.

The average age of Harvard students in 1810 was 15 1/2, the curriculum was primarily focused on Latin, Greek, math, and some sciences, and almost all of the students were white boys and young men from Boston families of means (excluding the men that attended the Harvard Indian School). They weren't reading anything unique but they were making connections. (Revolutionary generation: Harvard men and the consequences of independence by C. Wright is in-depth look at the connections that formed.) The young men created and joined networks that would lead to law firms, political alliances, and business opportunities. While networking happened at other colleges - including those in England - they lacked the panache of Harvard. One of America's oldest high schools was formed basically as a feeder school for the college - if you were a son from a New England family of means, Harvard was in your future from a young age.

However, there a variety of reasons a young man might not get in. If a particular class was full, the college could afford to turn away applicants to increase their prestige reputation. The president could also turn away a young man because they hadn't memorized the right Latin passage or failed to impress a particular member of the admissions testing panel. Pulling strings for a son or grandson was what one did if one wanted their progeny to have access to the name and networks.